Found problems: 85335
2008 ITest, 14
The sum of the two perfect cubes that are closest to $500$ is $343+512=855$. Find the sum of the two perfect cubes that are closest to $2008$.
2025 Malaysian IMO Training Camp, 4
For each positive integer $k$, find all positive integer $n$ such that there exists a permutation $a_1,\ldots,a_n$ of $1,2,\ldots,n$ satisfying $$a_1a_2\ldots a_i\equiv i^k \pmod n$$ for each $1\le i\le n$.
[i](Proposed by Tan Rui Xuen and Ivan Chan Guan Yu)[/i]
1990 IMO, 1
Chords $ AB$ and $ CD$ of a circle intersect at a point $ E$ inside the circle. Let $ M$ be an interior point of the segment $ EB$. The tangent line at $ E$ to the circle through $ D$, $ E$, and $ M$ intersects the lines $ BC$ and $ AC$ at $ F$ and $ G$, respectively. If
\[ \frac {AM}{AB} \equal{} t,
\]
find $\frac {EG}{EF}$ in terms of $ t$.
2004 Singapore MO Open, 3
Let $AD$ be the common chord of two circles $\Gamma_1$ and $\Gamma_2$. A line through $D$ intersects $\Gamma_1$ at $B$ and $\Gamma_2$ at $C$. Let $E$ be a point on the segment $AD$, different from $A$ and $D$. The line $CE$ intersect $\Gamma_1$ at $P$ and $Q$. The line $BE$ intersects $\Gamma_2$ at $M$ and $N$.
(i) Prove that $P,Q,M,N$ lie on the circumference of a circle $\Gamma_3$.
(ii) If the centre of $\Gamma_3$ is $O$, prove that $OD$ is perpendicular to $BC$.
1991 Arnold's Trivium, 93
Decompose the space of functions defined on the vertices of a cube into invariant subspaces irreducible with respect to the group of a) its symmetries, b) its rotations.
2006 Stanford Mathematics Tournament, 17
Car A is traveling at 20 miles per hour. Car B is 1 mile behind, following at 30 miles per hour. A fast fly can move at 40 miles per hour. The fly begins on the front bumper of car B, and flies back and forth between the two cars. How many miles will the fly travel before it is crushed in the collision?
2008 Bulgaria Team Selection Test, 2
In the triangle $ABC$, $AM$ is median, $M \in BC$, $BB_{1}$ and $CC_{1}$ are altitudes, $C_{1} \in AB$, $B_{1} \in AC$. The line through $A$ which is perpendicular to $AM$ cuts the lines $BB_{1}$ and $CC_{1}$ at points $E$ and $F$, respectively. Let $k$ be the circumcircle of $\triangle EFM$. Suppose also that $k_{1}$ and $k_{2}$ are circles touching both $EF$ and the arc $EF$ of $k$ which does not contain $M$. If $P$ and $Q$ are the points at which $k_{1}$ intersects $k_{2}$, prove that $P$, $Q$, and $M$ are collinear.
1979 IMO Shortlist, 13
Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$
2016 AIME Problems, 14
Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside of the plane of $\triangle ABC$ and are on the opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (The angle between the two planes). There is a point $O$ whose distance from each of $A,B,C,P$ and $Q$ is $d$. Find $d$.
1998 Putnam, 3
Let $H$ be the unit hemisphere $\{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}$, $C$ the unit circle $\{(x,y,0):x^2+y^2=1\}$, and $P$ the regular pentagon inscribed in $C$. Determine the surface area of that portion of $H$ lying over the planar region inside $P$, and write your answer in the form $A \sin\alpha + B \cos\beta$, where $A,B,\alpha,\beta$ are real numbers.
2007 Dutch Mathematical Olympiad, 2
Is it possible to partition the set $A = \{1, 2, 3, ... , 32, 33\}$ into eleven subsets that contain three integers each, such that for every one of these eleven subsets, one of the integers is equal to the sum of the other two? If so, give such a partition, if not, prove that such a partition cannot exist.
2000 Romania National Olympiad, 1
For the real numbers $a, b, c, d$, the following inequalities hold:
$$a + b + c \le 3d, \,\,\, b + c + d \le 3a, \,\,\,c + d + a \le 3b, \,\,\,d + a + b\le 3c.$$
Compare the numbers $a, b, c, d$.
2023 Sinapore MO Open, P3
Let $n \geq 2$ be a positive integer. For a positive integer $a$, let $Q_a(x)=x^n+ax$. Let $p$ be a prime and let $S_a=\{b | 0 \leq b \leq p-1, \exists c \in \mathbb {Z}, Q_a(c) \equiv b \pmod p \}$. Show that $\frac{1}{p-1}\sum_{a=1}^{p-1}|S_a|$ is an integer.
1985 AMC 12/AHSME, 27
Consider a sequence $ x_1, x_2, x_3, ...$, defined by
\begin{align*}x_1 &= \sqrt [3]{3}\\
x_2 &= \sqrt [3]{3} ^ {\sqrt [3]{3}},\end{align*}and in general
\[ x_n \equal{} (x_{n \minus{} 1}) ^ {\sqrt [3]{3}}\,\,\text{ for }\,\,n > 1.
\]What is the smallest value of $ n$ for which $ x_n$ is an integer?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 27$
MOAA Gunga Bowls, 2021.9
William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hours during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it take William to get to school and back.
[i]Proposed by William Yue[/i]
LMT Team Rounds 2021+, 5
In rectangle $ABCD$, $AB = 40$ and $AD = 30$. Let $C' $ be the reflection of $C$ over $BD$. Find the length of $AC'$.
2011 Turkey MO (2nd round), 4
$a_{1}=5$ and $a_{n+1}=a_{n}^{3}-2a_{n}^{2}+2$ for all $n\geq1$. $p$ is a prime such that $p=3(mod 4)$ and $p|a_{2011}+1$. Show that $p=3$.
2011 LMT, 8
There are four entrances into Hades. Hermes brings you through one of them and drops you off at the shore of the river Acheron where you wait in a group with five other souls, each of which had already come through one of the entrances as well, to get a ride across. In how many ways could the other five souls have come through the entrances such that exactly two of them came through the same entrance as you did? The order in which the souls came through
the entrances does not matter, and the entrance you went through is fixed.
2009 Miklós Schweitzer, 1
On every card of a deck of cards a regular 17-gon is displayed with all sides and diagonals, and the vertices are numbered from 1 through 17. On every card all edges (sides and diagonals) are colored with a color 1,2,...,105 such that the following property holds: for every 15 vertices of the 17-gon the 105 edges connecting these vertices are colored with different colors on at least one of the cards. What is the minimum number of cards in the deck?
2024 Junior Balkan Team Selection Tests - Romania, P2
Let $M$ be the midpoint of the side $AD$ of the square $ABCD.$ Consider the equilateral triangles $DFM{}$ and $BFE{}$ such that $F$ lies in the interior of $ABCD$ and the lines $EF$ and $BC$ are concurrent. Denote by $P{}$ the midpoint of $ME.$ Prove that"
[list=a]
[*]The point $P$ lies on the line $AC.$
[*]The halfline $PM$ is the bisector of the angle $APF.$
[/list]
[i]Adrian Bud[/i]
2013 Cuba MO, 2
Two equal isosceles triangles $ABC$ and $ADB$, with $C$ and $D$ located in different halfplanes with respect to the line $AB$, share the base $AB$. The midpoints of $AC$ and $BC$ are denoted by $E$ and $F$ respectively. Show that $DE$ and $DF$ divide $AB$ into three equal parts length.
1987 Poland - Second Round, 6
We assign to any quadrilateral $ ABCD $ the centers of the circles circumscribed in the triangles $ BCD $, $ CDA $, $ DAB $, $ ABC $. Prove that if the vertices of a convex quadrilateral $ Q $ do not lie on the circle, then
a) the four points assigned to quadrilateral Q in the above manner are the vertices of the convex quadrilateral. Let us denote this quadrilateral by $ t(Q) $,
b) the vertices of the quadrilateral $ t(Q) $ do not lie on the circle,
c) quadrilaterals $ Q $ and $ t(t(Q) $ are similar.
1996 May Olympiad, 5
You have a $10 \times 10$ grid. A "move" on the grid consists of moving $7$ squares to the right and $3$ squares down. In case of exiting by a line, it continues at the beginning (left) of the same line and in case of ending a column, it continues at the beginning of the same column (above). Where should we start so that after $1996$ moves we end up in a corner?
2009 Costa Rica - Final Round, 1
Let $ x$ and $ y$ positive real numbers such that $ (1\plus{}x)(1\plus{}y)\equal{}2$. Show that $ xy\plus{}\frac{1}{xy}\geq\ 6$
2024 ELMO Shortlist, N7
For a prime $p$, let $\mathbb{F}_p$ denote the integers modulo $p$, and let $\mathbb{F}_p[x]$ be the set of polynomials with coefficients in $\mathbb{F}_p$. Find all $p$ for which there exists a quartic polynomial $P(x) \in \mathbb{F}_p[x]$ such that for all integers $k$, there exists some integer $\ell$ such that $P(\ell) \equiv k \pmod p$. (Note that there are $p^4(p-1)$ quartic polynomials in $\mathbb{F}_p[x]$ in total.)
[i]Aprameya Tripathy[/i]